Sn

We begin our analyses with the sphere because it is both topologically and geometrically the simplest of
manifolds, and because its natural metric is such an important ingredient in so many other metrics.

An n-sphere of fixed radius r as embedded in Euclidean space of dimension n + 1 is the locus of
all points satisfying

r2 = Σ xi2

where the xi are coordinates in the
Cartesian chart.
It is important to realize that this definition is a convenient starting point, but that the sphere need
not be considered as an embedding in order to be defined. In an intrinsic definition, the sphere
is treated as a rotation of a (hyper-)hemisphere around a circle.
The standard metric is

ds2 = r2dΩ2,

where

dΩ2 =
dθ12 +
sin(θ1)2dθ22 +
sin(θ1)2
sin(θ2)2dθ32 + ... +

Π sin(θi)2dφ2

and r is simply a parameter specifying the curvature of the sphere.
We can use the following Mathematica code to generate this metric:

The first line creates a list whose elements are the variables t1, t2, etc., corresponding to
θ1, etc. The second line creates a list whose elements
are the diagonal entries in the final metric which is then created and filled in with the remaining
line.

We can see that c1 is a geodesic direction.
For n = 2, these are the "great circles" crossing the poles. Similarly, for n > 2 the c1 and
c2 directions define a geodesic hypersurface
and for n > 3 c1, c2 and c3 define a geodesic hypersurface.
In general, there n - 2 geodesic hypersurfaces defined by {c1 ... ci} for
i from 2 to n - 1.

Note also that when c1 is zero, c2 is a geodesic direction: the great circle around
the equator. In general, when c1 through ci are zero, ci+1 is a
geodesic direction.

Manifolds which admit metrics whose Ricci Tensors are a constant multiple of the metric are
called Einstein Manifolds. They are spaces of constant curvature and admit metrics which are
vacuum solutions
to Einstein's Equations with cosmological constant.
For Sn,

Λ = (n - 1) (n / 2 - 1) / r2

The invariants we have chosen to examine are

R

n (n - 1) / r2

Ra b Ra b

n (n - 1)2 / r4

Ra b c d Ra b c d

2 n (n - 1) / r4

Ra b Rca Rb c

n (n - 1)3 / r6

Ra b c d Ra c Rb d

n (n - 1)3 / r6

Ra b c d Rea Rb c d e

n (n - 1)2 / r6

Ra b c d Re fa b Rc d e f

4 n (n - 1) / r6

Ra b c d Reafc Rb e d f

n (n - 1) (n - 2) / r6

Ra b c d Reafc Rb f d e

n (n - 1) (n - 3) / r6

Ra b; c Ra b; c

0

Ra b; c Ra c; b

0

Ra b; a Rcb; c

0

Ra b c d; e Ra b c d; e

0

Ra b c d; a Reb c d; e

0

Euler class

2 / Area (Sn) for even n, otherwise 0

εa b c i ...
εe f gi ...
Rb c eh Rf g a h

- n (2 n - 4) (2 n - 2) / r4

Note that all of them are positive except the last, consistent with the knowledge that a sphere is
a space of positive curvature, and that the invariants involving
covariant derivatives are zero,
consistent with the knowledge that a sphere is a space of constant curvature. The overwhelming similarity
of these invariants is indicative of the geometrical simplicity of the sphere: they only depend on
the dimension and the parameter r.

The surface area of Sn is a factor in many computations. In general, it is